The minimum value of the sum of real numbers a⁻⁵, a⁻⁴, 3a⁻³, 1, a⁸ and a¹⁰…

Concept: For n positive real numbers, the AM–GM inequality says their arithmetic mean is always at least as large as their geometric mean (the nth root of their product), with equality exactly when all n numbers are equal.

Applying this here: Since a > 0, every term in a⁻⁵ + a⁻⁴ + 3a⁻³ + 1 + a⁸ + a¹⁰ is positive, so AM–GM can be applied once the terms are split into equal parts:

1. Since a > 0, every term in a⁻⁵ + a⁻⁴ + 3a⁻³ + 1 + a⁸ + a¹⁰ is positive, so AM–GM can be applied once the terms are split into equal parts.

2. Rewrite 3a⁻³ as three equal parts, a⁻³ + a⁻³ + a⁻³, turning the six given terms into eight positive terms: a⁻⁵, a⁻⁴, a⁻³, a⁻³, a⁻³, 1, a⁸, a¹⁰.

3. By AM–GM, the sum of these eight terms is at least 8 times the eighth root of their product.

4. Multiply the eight terms together to find the product: a⁻⁵ · a⁻⁴ · a⁻³ · a⁻³ · a⁻³ · 1 · a⁸ · a¹⁰ = a to the power (-5-4-3-3-3+8+10) = a⁰ = 1 (adding the exponents: -5-4-3-3-3+8+10 = 0).

5. Since the product is 1, its eighth root is also 1, so the sum is at least 8 × 1 = 8.

6. Equality in AM–GM holds only when all eight parts are equal; since one part is the constant 1, this forces a⁻⁵ = a⁻⁴ = a⁻³ = a⁸ = a¹⁰ = 1, which happens exactly when a = 1.

Cross-check: Substituting a = 1 back into the original sum confirms the bound is reached: 1 + 1 + 3 + 1 + 1 + 1 = 8. Because the sum decreases for a < 1 and increases for a > 1 (a single turning point at a = 1), this attained value of 8 is the true minimum for a > 0, not merely a lower bound.